A note on contests with a constrained choice set of effort

TL;DR

This paper analyzes a symmetric two-player contest with effort constraints, proving the existence and uniqueness of pure strategy Nash equilibrium, which closely resembles the unconstrained case unless players value prizes differently.

Contribution

It establishes the existence and uniqueness of pure strategy Nash equilibrium in constrained effort contests and explores how prize valuation differences affect equilibrium outcomes.

Findings

Unique pure strategy Nash equilibrium exists under standard assumptions.

Equilibria are close to the unconstrained case when players have similar prize evaluations.

Differences in prize valuation can lead to deviations from the unconstrained equilibrium.

Abstract

We consider a symmetric two-player contest, in which the choice set of effort is constrained. We apply a fundamental property of the payoff function to show that, under standard assumptions, there exists a unique Nash equilibrium in pure strategies. It is shown that all equilibria are near the unconstrained equilibrium. Perhaps surprisingly, this is not the case when players have different prize evaluations.